PART SECOND. SECTION I.-Of Planes, and of Bodies terminated by Planes. 195. IN the preceding sections, we have considered all the lines of any one of the figures which have been discussed, as in a plane surface ; that is, in a surface to which a straight line may be applied in every direction, so as to lie entirely in this surface. This has enabled us to unfold the elementary principles of linear geometry, and to measure the areas of surfaces bounded by straight lines or circular curves. We proceed now to investigate the geometrical relations in bodies. As the surfaces of many bodies consist, in whole or in part, of planes; and as all bodies involve the three dimensions of space, length, breadth and thickness; in discussing their geometrical character, it is necessary to consider not only the forms and magnitudes of planes, but their relative positions, their inclinations, and their relation to lines and points without them.* In discussing the general relations of lines to planes, and of planes to each other, we consider them as indefinitely extended, except where some limit is expressly stated. 196. If two straight lines are in the same plane, a second plane may be conceived to pass through one of these lines without coinciding with the other line ; it will therefore cut the first plane. But if any two points * In what follows, the figures though represented on a plane, are considered as embracing the three dimenision of space. And where any line is supposed to pass behind any part of a body or of a plane, it is indicated in the figure by a dotted line. common to the two planes, be taken, a straight line drawn between them will lie wholly in each of the planes (8); therefore—The interscetion of two planes is a straight line. 197. If the second plane be turned about the straight line through which it passes, till it coincides with the other line, it must also coincide with the other plane : Two straight lines, therefore, are sufficient to determine the position of a plane : And three points, not in the same straight line, will also determine the position of a plane; for the plane may be conceived to revolve about a straight line passing through two of the points; and if it be placed so as to coincide with the other point, it can revolve no farther without leaving it; it is therefore fixed. It follows from this that any two straight lines which cut each other, are in the same plane; for a plane may be passed through one of them, as AB (fig. 108), and Fig.108. turned till it coincide with the other CD. It is evident, therefore, that—any three points, not in the same straight line, being joined, two and two, by three straight lines, a plane triangle will be formed ; if four points are so joined, the quadrilateral, thus formed, may be a plane figure ; but if the plane passing through three of the points, do not at the same time pass through the fourth, the quadrilateral cannot be considered as a single surface. 19s. A straight line is said to be parallel to a plane, when it does not incline towards the plane, in either direction. A line parallel to a plane will therefore be, in every part, at the same distance from the plane; and consequently can never meet it, however far produced. A straight line is perpendicular to a plane when its inclination to the plane is the same on all sides. It is evident that the straight line will, in this case, be perpendicular to every straight line drawn in this plane, through the foot of the perpendicular. 199. , Let us suppose a straight line PD (fig. 109) Fig.109. to revolve about the straight line AB, always perpendicular to it, AB being considered as fixed. It is evident that PD, by this revolution, will generate a plane surface; for in whatever position the revolving line be placed, in the position PE, for instance, a plane may be conceived to pass through EPA, which being extended 7 Fig.109. Fig.110. will also pass through the opposite position of the revolv- Fig.114. Fig.115. tion, and the angle which these two lines make with each |